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Symmetry 2010, 2(2), 707-721; doi:10.3390/sym2020707

Symmetry and Asymmetry Level Measures

Departamento de Matemáticas Fundamentales, Facultad de Ciencias de la UNED, Paseo Senda del Rey, 9. 28040 Madrid, Spain
Received: 23 November 2009 / Revised: 30 March 2010 / Accepted: 6 April 2010 / Published: 8 April 2010
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
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Usually, Symmetry and Asymmetry are considered as two opposite sides of a coin: an object is either totally symmetric, or totally asymmetric, relative to pattern objects. Intermediate situations of partial symmetry or partial asymmetry are not considered. But this dichotomy on the classification lacks of a necessary and realistic gradation. For this reason, it is convenient to introduce "shade regions", modulating the degree of Symmetry (a fuzzy concept). Here, we will analyze the Asymmetry problem by successive attempts of description and by the introduction of the Asymmetry Level Function, as a new Normal Fuzzy Measure. Our results (both Theorems and Corollaries) suppose to be some new and original contributions to such very active and interesting field of research. Previously, we proceed to the analysis of the state of art.
Keywords: Fuzzy Analysis; Generalized Fuzzy Measures; Entropy; Specificity; Symmetry; Anti-symmetry Fuzzy Analysis; Generalized Fuzzy Measures; Entropy; Specificity; Symmetry; Anti-symmetry
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Garrido, A. Symmetry and Asymmetry Level Measures. Symmetry 2010, 2, 707-721.

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